Calculus/Limits/Exercises

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Limits/Exercises

Basic Limit ExercisesEdit

1.  
 
 
2.  
 
 
3.  
 
 
4.  
 
 

Solutions

One-Sided LimitsEdit

Evaluate the following limits or state that the limit does not exist.

5.  
 
 
6.  
 
 
7.  
 
 
8.  
The limit does not exist.
The limit does not exist.
9.  
 
 
10.  
 
 

Solutions

Two-Sided LimitsEdit

Evaluate the following limits or state that the limit does not exist.

11.  
 
 
12.  
The limit does not exist.
The limit does not exist.
13.  
The limit does not exist.
The limit does not exist.
14.  
 
 
15.  
 
 
16.  
 
 
17.  
 
 
18.  
 
 
19.  
 
 
20.  
The limit does not exist.
The limit does not exist.
21.  
 
 
22.  
The limit does not exist.
The limit does not exist.
23.  
 
 
24.  
 
 
25.  
The limit does not exist.
The limit does not exist.
26.  
 
 
27.  
 
 
28.  
 
 
29.  
 
 
30.  
 
 
31.  
The limit does not exist.
The limit does not exist.
32.  
The limit does not exist.
The limit does not exist.
33.  
The limit does not exist.
The limit does not exist.

Solutions

Limits to InfinityEdit

Evaluate the following limits or state that the limit does not exist.

34.  
 
 
35.  
 
 
36.  
 
 
37.  
 
 
38.  
 
 
39.  
 
 
40.  
 
 
41.  
 
 
42.  
 
 
43.  
 
 
44.  
 
 
45.  
 
 
46.  
 
 

Solutions

Limits of Piecewise FunctionsEdit

Evaluate the following limits or state that the limit does not exist.

48. Consider the function

 
a.  
 
 
b.  
 
 
c.  
The limit does not exist
The limit does not exist

49. Consider the function

 
a.  
 
 
b.  
 
 
c.  
 
 
d.  
 
 
e.  
 
 
f.  
 
 

50. Consider the function

 
a.  
 
 
b.  
 
 
c.  
 
 
d.  
 
 

Solutions

Intermediate Value TheoremEdit

51. Use the intermediate value theorem to show that there exists a value   for   from  . If you cannot use the intermediate value theorem to show this, explain why.
Notice   is continuous from  . Ergo, the intermediate value theorem applies. For all  , there exists a   so that  .  
Notice   is continuous from  . Ergo, the intermediate value theorem applies. For all  , there exists a   so that  .  
52. Use the intermediate value theorem to show that there exists an   so that   for   from  . If you cannot use the intermediate value theorem to show this, explain why.
Notice   is continuous from  . Ergo, the intermediate value theorem applies.
 
 

It is known the following is true:  . From there, we can directly argue the following:

 
By the intermediate value theorem, if   is continuous from  , then there exists an   so that   for  .  
Notice   is continuous from  . Ergo, the intermediate value theorem applies.
 
 

It is known the following is true:  . From there, we can directly argue the following:

 
By the intermediate value theorem, if   is continuous from  , then there exists an   so that   for  .  
53. Use the intermediate value theorem to show that there exists a value   so that   for   from  . If you cannot use the intermediate value theorem to show this, explain why.
Notice   is not continuous for   since   is unbounded. Ergo, the intermediate value theorem cannot be used to solve this problem.
Notice   is not continuous for   since   is unbounded. Ergo, the intermediate value theorem cannot be used to solve this problem.

Solutions

External LinksEdit


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Limits/Exercises